The number of integer points on Vinogradov's quadric (Q987409)

Let \(V^+(N)\) denote the number of integer solutions to the diophantine system \[ x_1^2+x_2^2+x_3^2=y_1^2+y_2^2+y_3^2,\quad x_1+x_2+x_3=y_1+y_2+y_3 \tag{1} \] with \(1\leq x_j,y_j\leq N\). \textit{N. N. Rogovskaya} [Diophantine approximations, 2, Collect. Artic., Moskva 1986, 78--84 (1986; Zbl 0648.10010)], improving work of Bykovskiĭ, obtained the asymptotic formula \(V^+(N) = \frac{18}{\pi^2} N^3 \log N + O(N^3)\) by an elementary argument. Here, the authors prove a more precise approximation to \(V^+(N)\), namely \[ V^+(N)=\frac{18}{\pi^2}N^3\log N +\frac{3}{\pi^2}\left(12\gamma-6\frac{\zeta'(2)}{\zeta(2)}-5\right)N^3+O(N^{\frac 52}\log N), \] where \(\gamma\) is Euler's constant and \(\zeta\) the Riemann zeta-function. The variety defined by (1) is the simplest one related to Vinogradov's mean value theorem, and therefore the authors refer to it as Vinogradov's quadric. Their motivation to study Vinogradov's quadric arose from two families of paucity theorems. The first of these is connected with Vinogradov's mean value theorem. Alternatively, the quadric (1) arises as the special case \(k = 2\) in the resp. series of pairs of equations.